Using Euler's equations in Cartesian form, or otherwise, show that under gravitational force alone the free surface of a fluid in solid body rotation with

$u=(-$yhat$(x)+$xhat$(y)),$

and hat$(z)$ vertically upwards, is a paraboloid.

A deep cylindrical bucket whose axis is vertical is of radius a and contains a layer of water initially at rest and of constant depth $h.$ The bucket is rotated about its axis at angular velocity $$ until the fluid is in solid body rotation. For what value of $$ is precisely half the area of the bottom of the bucket dry?